2019 TJC P1 Q5

In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system such as a lens.

In the diagram below, a light ray parallel to the horizontal axis is reflected at point \(B\) on the circular lens centred at point \(C\) and has radius \(r\) cm. Let \(\angle BCF=\theta \) radians. \(FM\) is the perpendicular bisector of \(CB\).

1. Show that \(CF=\frac{r}{k\cos \theta }\), where \(k\) is a real constant to be determined.[1]
2. Hence find the series expansion for \(CF\) if \(\theta \) is sufficiently small for \({{\theta }^{3}}\) and terms in higher powers of \(\theta \) to be neglected.[2]

Suppose that the source of the light ray is now repositioned such that \(\angle BCF=\left( \theta +\frac{\pi }{6} \right)\) radians.

3. Find the corresponding series expansion for \(CF\), up to and including the term in \({{\theta }^{2}}\).[4]